So earlier today at lunch someone brought up the fact that Nadal has won 9 out of the last 10 Roland Garros tournaments. For some reason I started thinking that no matter when he lost that one tournament, he must have won at least 5 of those tournaments 5 years in a row.
More generally, if someone loses $k$ out of $n$ games, then they must have won at least $\lceil \frac{n-k}{k+1} \rceil$ games in a row (it seems pretty obvious although I can't prove it, maybe using stars and bars?).
Is there a name for this "minimum" number?
I also know that this number also represents the maximum number of games won in a row, when we try to divide the games into "as similar as possible" winning streaks